波士顿房价的三种预测方式（模型预测，最小二乘法，多元线性回归）

首先导入库并避免显示错误FutureWorning

import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import seaborn as sns
from sklearn.datasets import load_boston
from sklearn.model_selection import train_test_split
from sklearn.linear_model import LinearRegression
from sklearn import metrics
from sklearn import preprocessing
import warnings

warnings.filterwarnings("ignore")  # 避免显示FutureWorning

预处理数据并筛选相关系数较大的特征值
注释的第一部分：为显示各个影响因素与价格的关系图
注释的第二部分：通过一元线性回归，将筛选的三个相关系数最大的特征值与价格的关系图绘制出来

boston = load_boston()
x = boston['data']  # 所有特征信息
y = boston['target']  # 房价
feature_names = boston['feature_names']  # 十三个影响因素
boston_data = pd.DataFrame(x, columns=feature_names)  # 特征信息和影响因素
boston_data['price'] = y  # 添加价格列
'''
for i in range(13):
    plt.figure(figsize=(10, 7))
    plt.scatter(x[:,i],y,s=2)#某影响因素的特征信息与房价
    plt.title(feature_names[i])
plt.show()'''
cor = boston_data.corr()['price']  # 求取相关系数
print(cor)
# 取相关系数大于0.5的特征值 RM LSTAT PTRATIO price

boston_data = boston_data[['LSTAT', 'PTRATIO', 'RM', 'price']]  # 保留该四组因素
y = np.array(boston_data['price'])
feature_names = ['LSTAT', 'PTRATIO', 'RM', 'price']
'''
for i in range(3):
    x = np.array(boston_data[feature_names[i]])
    var = np.var(x,ddof=1)
    cov = np.cov(x,y)[0][1]
    k = cov/var
    b = np.mean(y) - k*np.mean(x)
    y_price = k*x + b
    plt.plot(x,y,'b.')
    plt.plot(x,y_price,'k-')
    plt.show()'''

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使用模型求解

通过LinearRegression库直接实现

boston_data = boston_data.drop(['price'], axis=1)
x = np.array(boston_data)
print(x)
train_x, test_x, train_y, test_y = train_test_split(x, y, test_size=0.3, random_state=0)  # 分割训练集和测试集

# 直接使用模型求解
# 加载模型
li = LinearRegression()
# 拟合数据
li.fit(train_x, train_y)
# 进行预测
y_predict = li.predict(test_x)
plt.figure(figsize=(10, 7))
plt.plot(y_predict, 'b-')
plt.plot(test_y, 'r--')
plt.legend(['predict', 'true'])
plt.title('Module')
plt.show()
mes = metrics.mean_squared_error(y_predict, test_y)
print(mes)

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最小二乘法求解

对系数B矩阵求偏导，偏导为0时，即为目标解

boston_data.insert(loc=0, column='one', value=1)
print(boston_data)
x = np.array(boston_data)
train_x, test_x, train_y, test_y = train_test_split(x, y, test_size=0.3, random_state=0)  # 分割训练集和测试集
xT = x.T
x_mat = np.mat(train_x)
y_mat = np.mat(train_y).T
xT = x_mat.T
B = (xT * x_mat).I * xT * y_mat
print(B)
y_predict = test_x * B
plt2 = plt.figure(figsize=(10, 7))
plt2 = plt.plot(y_predict, 'b-')
plt2 = plt.plot(test_y, 'r--')
plt2 = plt.legend(['predict', 'true'])
plt2 = plt.title('Least Sqaure Method')
plt.show()
# [[22.17686885]
# [-0.55760828]
# [-1.11165635]
# [ 4.44512502]]

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多元线性回归

简单多元线性回归（梯度下降算法与矩阵法）
机器学习方法之线性回归(LR)

首先处理x中的数据，在x每一行前添加1

print(boston_data)
x = np.array(boston_data)
train_x, test_x, train_y, test_y = train_test_split(x, y, test_size=0.3, random_state=0)  # 分割训练集和测试集
xT = x.T
print(xT)
x_mat = np.mat(train_x)
y_mat = np.mat(train_y).T
xT = x_mat.T
B = [[22.1768],[-0.5576],[-1.11165],[4.44512]]
B = np.mat(B)
Bp = -2*xT*(y_mat-x_mat*B) # 即B的偏导
while 1:
    B = B - 0.000005*Bp
    Bp = -2 * xT * (y_mat - x_mat * B)
    for i in range(4):
        sum = 0
        sum += abs(Bp[i])
    print(sum)
    if sum <=0.00001: break
    print(B)
print("最终B：",B)
# 最终B： [[22.17682455]
#   [-0.55760809]
#  [-1.11165553]
#  [ 4.44512921]]
y_predict = test_x*B
plt3 = plt.figure(figsize=(10, 7))
plt3 = plt.plot(y_predict,'b-')
plt3 = plt.plot(test_y,'r--')
plt3 = plt.legend(['predict', 'true'])
plt.title('Multiple Linear Regression')
plt.show()